Attachment-induced ionization instability has been experimentally observed in O-2 and CF4 capacitive RF discharges using time-resolved voltage probe, Langmuir probe, optical emission and mass spectrometry measurements. This instability occurs under specific conditions of power and pressure, and produces synchronized oscillations in the kilohertz range on potentials, emission intensity and positive ion fluxes. In contrast, the SF6 plasma was observed to remain stable under all experimental conditions. This can be understood by considering attachment and ionization cross sections of these gases and applying the theoretical criterion of instability. Contrary to O-2 and CF4, the attachment rate coefficient of SF6 is very high at low energy and has a negative dependence on the electronic temperature. The application of the criterion shows clearly that O-2 and CF4 plasmas are unstable at low electronic temperature, and that the SF6 plasma is stable due to its particular low-energy attachment cross section.